B-site ordered A$_2$B'B''O$_6$ double perovskites have a variety of applications as magnetic materials. Here we show that diamagnetic $d^{10}$ and $d^0$ B'' cations have a significant effect on the magnetic interactions in these materials. We present a neutron scattering and theoretical study of the Mn$^{2+}$ double perovskite Ba$_2$MnTeO$_6$ with a $4d^{10}$ Te$^{6+}$ cation on the B''-site. It is found to be a Type I antiferromagnet with a dominant nearest-neighbor $J_1$ interaction. In contrast, the $5d^0$ W$^{6+}$ analogue Ba$_2$MnWO$_6$ is a Type II antiferromagnet with a significant next-nearest-neighbor $J_2$ interaction. This is due to a $d^{10}$/$d^0$ effect, where the different orbital hybridization with oxygen 2p results in different superexchange pathways. We show that $d^{10}$ B'' cations promote nearest neighbor and $d^0$ cations promote next-nearest-neighbor interactions. The $d^{10}$/$d^0$ effect could be used to tune magnetic interactions in double perovskites.